A diffusion model for the Dirac equation

V. Balakrishnan, S. Lakshmibala

The master equation for dichotomous diffusion (DD) (the integral of a random telegraph process) is the well-known telegrapher's equation, which is converted to the Klein–Gordon equation by a simple transformation. After a brief recapitulation of the solution and of the analogy between DD and the Dirac equation in one spatial dimension, we consider velocity -biased DD. The corresponding master equation and its solution are presented. It is shown that these may be interpreted physically in terms of a Lorentz transformation to a frame moving with a boost velocity equal to the mean drift velocity of the diffusing particle. The modifications that arise in the connection with the Dirac equation are also exhibited. The correspondence between the rest mass of the Dirac particle and the frequency of direction reversal in the DD is shown to be modified precisely by the time dilatation correction to the latter quantity.

UMBERTO LOMBARDO, G. Giuliani, Yifei Niu

Heinz‐Peter Breuer, Francesco Petruccione

A relativistic generalization of the quantum-state diffusion model is developed. The model describes a Dirac electron which is coupled to an external electromagnetic field and a dissipative environment. A relativistically covariant stochastic Dirac equation is obtained by regarding the state vector as a functional on a certain set of spacelike hypersurfaces in Minkowski space and by the definition of an appropriate Hilbert bundle on this set. The integrability condition of the stochastic process and the corresponding covariant density matrix equation are derived. Further, the relativistic equations governing the dynamical state-vector localization are deduced.

Morton D. Kostin

According to Schrödinger–Fuerth quantum diffusion theory the equations of quantum theory can be derived by applying the Fuerth transformation to the equations of diffusion theory. By introducing a complex flux whose real and imaginary parts are the x and y components of the flux, a complex form of the two-dimensional diffusion equation is obtained and the equation is then generalized to three dimensions. Application of the Fuerth transformation to the generalized complex diffusion equation yields a quantum equation, which is shown to be equivalent to the Dirac equation.

Theodore Frankel